The area of the region, expressed as a definite integral is?

Question

Consider the region bounded by the graphs of f(x)=x^2-3 and g(x)=2x.

Answer 1

First graph the 2 functions on a graphing calculator or on wolframalpha to see that g(x) is above ƒ(x)

Then you find what x values the two functions intersect at to determine the limits on the definite integral

My favorite way to do this is to figure out what are the x values when the y value of the 2 functions are equal

so y=x²-3 is equal to y=2x

x²-3=2x solve for x

x²-2x-3=0

(x-3)(x+1)=0

x-3=0 and x+1=0 yields x=-1 and x=3

Area between 2 functions= ∫(top function -bottom function)dx

∫(2x-(x²-3))dx from -1 to 3

answer is

∫(2x-x²+3))dx from -1 to 3